We investigate an XY spin-glass model in which both spins and interactions (or couplings) evolve in time, but with widely separated time-scales. For large times this model can be solved using replica theory, requiring two levels of replicas, one level for the spins and one for the couplings. We define the relevant order parameters, and derive a phase diagram in the replica-symmetric approximation, which exhibits two distinct spin-glass phases. The first phase is characterized by freezing of the spins only, whereas in the second phase both spins and couplings are frozen. A detailed stability analysis leads also to two distinct corresponding de Almeida-Thouless lines, each marking continuous replica-symmetry breaking. Numerical simulations support our theoretical study.

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Abstract. We investigate an XY spin-glass model in which both spins and couplings evolve in time: the spins change rapidly according to Glauber-type rules, whereas the couplings evolve slowly with a dynamics involving spin correlations and Gaussian disorder. For large times the model can be solved using replica theory. In contrast to the XY-model with static disordered couplings, solving the present model requires two levels of replicas, one for the spins and one for the couplings. Relevant order parameters are defined and a phase diagram is obtained upon making the replica-symmetric Ansatz. The system exhibits two different spin-glass phases, with distinct de AlmeidaThouless lines, marking continuous replica-symmetry breaking: one describing freezing of the spins only, and one describing freezing of both spins and couplings.Recently various models with a coupled dynamics of fast Ising spins and slow couplings have been studied (see e.g. [1,5] and references therein). In addition to physical motivations, such as understanding the simultaneous learning and retrieval in recurrent neural networks or the influence of slow atomic diffusion processes in disordered magnetic systems, there is a more theoretical interest in such models in that they generate the replica formalism for a finite number of replicas n. Moreover, the replica number is found to have a physical meaning as the ratio of two temperatures (characterizing the stochasticity in the spin dynamics and the coupling dynamics, respectively). In this letter we extend the methods and results obtained for Ising spin models to a classical XY spin-glass with dynamic couplings, whose spin variables are physically more realistic than Ising ones. In addition, the XY model is closely related to neural network models of coupled oscillators, which provide a phenomenological description of neuronal firing synchronization in brain tissue. We solve our model upon making the replica-symmetric Ansatz, and calculate the de Almeida-Thouless (AT) lines [6] (of which here there are two types), where continuous transitions occur to phases with broken replica symmetry. In doing so we also improve the calculations of [3]. As §

The parallel dynamics of extremely diluted symmetric Q-Ising neural networks is studied for arbitrary Q using a probabilistic approach. In spite of the extremely diluted architecture the feedback correlations arising from the symmetry prevent a closed-form solution in contrast with the extremely diluted asymmetric model. A recursive scheme is found determining the complete time evolution of the order parameters taking into account all feedback. It is based upon the evolution of the distribution of the local field, as in the fully connected model. As an illustrative example an explicit analysis is carried out for the Q = 2 and Q = 3 model. These results agree with and extend the partial results existing for Q = 2. For Q > 2 the analysis is entirely new. Finally, equilibrium fixed-point equations are derived and a capacity-gain function diagram is obtained.

Abstract. The parallel dynamics of the asymmetric extremely diluted Ashkin-Teller neural network is studied using signal-to-noise analysis techniques. Evolution equations for the order parameters are derived, both at zero and finite temperature. The retrieval properties of the network are discussed in terms of the four-spin coupling strength and the temperature. It is shown that the presence of a four-spin coupling enhances the retrieval quality.PACS. 64.60.Cn Order-disorder transformations; statistical mechanics of model systems -75.10.Hk Classical spin models -87.10.+e General, theoretical, and mathematical biophysics -02.50.-r Probability theory, stochastic processes and statistics